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  "body": "\n## Statement\n\nLet $\\mu \\colon \\C \\to \\C$ be a measurable function with $\\norm{\\mu}_\\infty \u003c 1$.\nThen there is a unique [[quasiconformal|SB8KG46NHP]] homeomorphism $f \\colon \\C \\to \\C$ such that:\n\n1. $f$ solves the [[Beltrami equation|9J7GFJ1GPF]] $f_{\\bar z} = \\mu f_z$ almost everywhere on $\\C$.\n2. $f(0) = 0$ and $f(1) = 1$.\n\n## Remarks\n\nWhen $\\mu$ has compact support, one looks for $f$ of the form $f(z) = z + \\frac{1}{\\pi}\\int_\\C \\frac{h(w)}{z - w}\\, dA(w)$, where $h = f_{\\bar z}$.\nThe Beltrami equation then becomes $h = \\mu + \\mu S h$, where $S$ is the [[Beurling–Ahlfors transform|TX32GDQ330]].\nThe operator $S$ is an isometry of $L^2$, and its norm on $L^p$ is close to $1$ for $p$ close to $2$.\nSo the equation has a solution given by a Neumann series in $L^p$ for some $p \u003e 2$.\nThe general case follows by composing solutions.\nUniqueness holds because two normalized solutions differ by a $1$-quasiconformal, hence conformal, self-map of $\\C$ fixing $0$ and $1$, which is the identity.\nThe theorem also gives analytic dependence of $f$ on $\\mu$, a fact that is central in Teichmüller theory and holomorphic dynamics.\nThe condition $\\norm{\\mu}_\\infty \u003c 1$ cannot be weakened to $\\abs{\\mu} \u003c 1$ almost everywhere: for some such $\\mu$ no homeomorphic solution exists.\n\n## Explanations\n\n### L1 human\n\nImagine that at each point of the plane someone draws a tiny ellipse, with the only rule that the ellipses are never thinner than a fixed ratio.\nThe theorem says that there is a way to deform the plane so that all these ellipses become circles.\nThe deformation is unique once two points are pinned down.\nSo any such field of ellipses is just the round structure of the plane seen through a distorted map.\n\n### L2 human\n\nThe theorem is the main existence result for [[quasiconformal mappings|SB8KG46NHP]].\nIt turns questions about conformal structures into questions about bounded measurable functions, which can be changed freely.\nIn complex dynamics it is used in quasiconformal surgery: one builds an invariant ellipse field and then straightens it to get a new holomorphic map.\nIn Teichmüller theory it shows that Beltrami coefficients parametrize deformations of Riemann surfaces, and the holomorphic dependence on $\\mu$ gives the complex structure of [[Teichmüller space|ZZGC6BZNJ5]].\nFor $\\mu = 0$ the solution is the identity, so the statement extends the uniqueness of conformal automorphisms of $\\C$ fixing two points.\n",
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